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COB298 FORMULAÇÃO BIDIMENSIONAL GERAL PARA PROBLEMAS INVERSOS DE CONDUÇÃO DE CALOR / A GENERAL TWO-DIMENSIONAL FORMULATION FOR INVERSE HEAT CONDUCTION PROBLEMS

J. P. Alencar Jr. and H. R. B. Orlande

DEM/PEM-EE/COPPE - UFRJ, Cid. Universitária, Cx. Postal: 68503, Rio de Janeiro, RJ, 21945-970, Brasil

M. N. Ozisik

Dept. of Mechanical Engineering. - NCSU, PO. Box: 7910, Raleigh, NC 27695-7910, USA

In this paper we present a general solution for two-dimensional boundary inverse heat conduction problems, by using the conjugate gradient method of minimization together with an elliptic scheme of numerical grid generation. The direct problem, as well as other auxiliary problems, are formulated in terms of generalized coordinates in a computational domain, where they are solved by finite-differences over a rectangular region. Simulated measurements are used to illustrate the application of the present approach with a practical inverse problem of engineering interest.

Keywords: Boundary Inverse Problem, Conjugate Gradient Method, Function Estimation, Numerical Grid Generation, Generalized Coordinates.

COB319 SCYL: UM PROGRAMA PARA A SOLUÇÃO NUMÉRICA EM PARALELO DAS EQUAÇÕES DE NAVIER-STOKES / SCYL: A PROGRAM FOR THE NUMERICAL SOLUTION IN PARALLEL OF THE NAVIER-STOKES EQUATIONS

Armando de Oliveira Fortuna

Departamento de Ciências de Computação e Estatística - ICMSC - USP

Av. Dr. Carlos Botelho, 1465-CP 668-CEP 13560-970-São Carlos, SP, Brasil-Email: fortuna@icmsc.sc.usp.br

The use of parallel computing in Computational Fluid Dynamics problems has greatly increased over the past few years, placing an important and useful tool in the hands of scientists and engineers. This works discusses the implementation of a solution algorithm for the incompressible Navier-Stokes equations using parallel computation techniques. Program SCYL is written in FORTRAN 77 and utilizes the message passing library PVM to distribute and control computations between different computers. After discretization the resulting system of algebraic equations is solved using an iterative procedure. Convergence is acceleratedwith the use of multigrid. The parallelized code running on a cluster of Pentiumä processor machines displays good speed-up when compared to the serial version.

Keywords: Incompressible flow, Navier-Stokes equations, multigrid, parallel computers, PVM

Escoamentos incompressíveis, Equações de Navier-Stokes, Multigrid, Computadores Paralelos, PVM

COB347 APLICAÇÃO DE TÉCNICAS MULTIGRID EM PROBLEMAS DE DINÂMICA DOS FLUIDOS/APPLICATIONS OF MULTIGRID METHODS TO FLUID DYNAMICS PROBLEMS

Rivânia H. Paulino & Washington Braga

Departamento de Engenharia Mecânica, PUC-Rio, R. Marquês de São Vicente, 225, CEP: 22453-900, Rio de Janeiro, RJ,

E-mail: wbraga@mec.puc-rio.br

The present work deals with the numerical solution of the Navier-Stokes equations, written in the stream function-vorticity form, by the finite difference method and acceleration techniques using multiple meshes. The classical method (storage of the correction) and the method FAS (Full Approximation Storage) have been tested. The results obtained clearly show that a very efficient computational scheme has been achieved with the multigrid method. For example, when comparing this method with the basic SOR method, relative gains in processing time in the order of 80% have been obtained.

Keywords: Solução Numérica, Navier-Stokes, Multigrid, Full Approximation Storage/

Numerical Solution, Navier-Stokes, Multigrid, Full Approximation Storage.

COB755 APPLICATION OF THE UNIFIED FINITE APPROACH EXPONENTIAL SCHEME TO THE CAVITY PROBLEM

Dr. José Ricardo Figueiredo

State University at Campinas, FEM, DE, 13083-970, Campinas, São Paulo, Brazil

This paper presents the application of a new discretization scheme for convective-diffusive fluid transport equations to the Navier-Stokes equations. The scheme is a conservative-form discretization constructed within the control-volume approach employing an exponential interpolating curve obtained as the exact solution of an approximated equation that admits a source term, which is computed by incorporating Allen’s finite difference approach into the finite volume method. It has been called the Unified Finite Approach Exponential Scheme, UNIFAES. A simple one-dimensional test case shows the scheme’s ability to cope with irregular grids The main test problem considered is the recirculating flow in a cavity due to a moving wall, where UNIFAES demonstrates accuracy superior to the conservative form schemes central differencing and simple exponential scheme.

Keywords: CFD, Finite Volume, Finite Difference, UNIFAES, cavity problem

COB764 A TIME SPLITTING METHOD FOR EVOLUTION EQUATIONS MODELLING BENARD CONVECTION / UM MÉTODO DE SEPARAÇÃO TEMPORAL DE VARIÁVEIS PARA EQUAÇÕES DE EVOLUÇÃO UTILIZADAS NA MODELAGEM DE CONVECÇÃO DE BENARD

J. Pontes

PEMM/COPPE/UFRJ, PO Box 68505, 21945--970 Rio de Janeiro, R. J. Brazil jopontes@metalmat.ufrj.br

C. I. Christov

National Intitute of Meteorology and Hydrology, 66 Tsarigradsko Chaussee, Sofia 1784, Bulgaria christov@meteo.bg

Pattern formation in a thin layer of fluid heated by below is often studied by numerical integration of nonlinear parabolic equations, containing fourth-order space-derivatives in the x and the y directions. A finite-difference semi-implicit coordinate-splitting scheme of first order in time and of second order in space is developed and applied to solving two such equations in finite geometries, namely the Swift- Hohenberg and the Knobloch equations. The construction of the scheme is accomplished by assigning terms to the implicit and to the explicit parts, and by replacing the spatial and time operators by discrete representations. The stability properties of the algorithm are defined at this stage. The scheme is splitted in two equivalent equations, to reduce the storage requirements. The resulting algebraic linear system displays a left hand side operator with a pentadiagonal structure and is solved by gaussian elimination with pivoting. Some results and a criteria to characterize the asymptotic state are presented.

Keywords: Nonlinear Systems, Bénard Convection, Implicit Methods, Finite Difference Methods

Sistemas não-lineares, Convecção de B'enard, M'etodos Impl'icitos, M'etodo de diferen¸cas finitas

COB776 CELL-IMPLICIT NUMERICAL COMPUTATION OF FLOW FIELD AND HEAT TRANSFER IN INCLINED CAVITIES

Marcelo J.S. de Lemos

Departamento de Energia, Instituto Tecnológico de Aeronáutica-CTA

12228-900 - São José dos Campos, SP, Brasil - E-mail: mlemos@tecsat.com.br

This paper reports numerical results for incompressible thermally-driven flows in tilted cavities obtained with a point-wise solution scheme. The discretized momentum equations are applied to each computational cell and then, together with the mass-continuity and energy equations, are solved directly for each grid node. The effect of cavity inclination on the thermal field is discussed upon. Flow pattern and mass residual behavior are also reported.

Keywords: Tilted Cavities, CFD, Numerical Methods, Laminar Flow

COB786 Comparação entre Vários Algoritmos de Fatoração Aproximada na Solução das Equações de Navier-Stokes / COMPARISON OF SOME APPROXIMATE FACTORIZATION ALGORITHMS IN THE SOLUTION OF THE NAVIER-STOKES EQUATIONS

Edisson S. G. Maciel⁽¹⁾ & João L. F. Azevedo⁽²⁾

⁽¹⁾ Instituto Tecnológico de Aeronáutica, Centro Técnico Aerospacial - CTA/ITA - mesg@aer.ita.cta.br

⁽²⁾ Instituto de Aeronáutica e Espaço, Centro Técnico Aerospacial - CTA/IAE - azevedo@ase2.iae.cta.br

12228-904 - São José dos Campos - SP - Brasil

The Navier-Stokes equations, written in conservative form, are applied to simulate the flowfield in a two-dimensional convergent-divergent transonic nozzle. A finite difference formulation is used to perform the spatial discretization in a generalized coordinate system. The march in pseudo-time is performed by the implicit Euler method to obtain steady state solutions. Three implicit, approximately factored, difference schemes are described and tested in the present work. These include the standard Beam and Warming algorithm, Chaussee and Pulliam´s diagonal version of it, and the Steger and Warming flux vector splitting scheme. Several options for spatial discretization of the split fluxes are studied in connection with the Steger and Warming scheme. The main objectives of the present work are to perform a comparative study of the different algorithms and to assess their cost/accuracy performance characteristics.

Keywords: Equações de Navier-Stokes, Escoamento em Bocal, Esquemas Implícitos, Esquemas de Diferenças Centradas, Separação de Vetores de Fluxo / Navier-Stokes Equations, Nozzle Flow, Implicit Schemes, Central Difference Schemes, Flux Vector Splitting.

COB794 ANÁLISE DO ESFORÇO COMPUTACIONAL DE SOLUÇÕES MULTIGRID DE PROBLEMA CONDUTIVO-CONVECTIVO / COMPUTATIONAL EFFORT ANALYSIS OF MULTIGRID SOLUTIONS OF CONDUCTIVE-CONVECTIVE PROBLEM

José Antonio Rabi & Marcelo José Santos de Lemos

Depto. Engenharia Mecânica-Aeronáutica, Instituto Tecnológico de Aeronáutica - ITA / CTA

12.228-901 S.J.Campos SP Brasil - E-mails: jrabi@mec.ita.cta.br, lemos@mec.ita.cta.br

Multigrid methods have been used for predicting flows achieving considerable computational time savings. In the present paper, a multigrid method has been applied to a finite-volume numerical solution of a 2-D conductive-convective problem. Structured and regular grids of different sizes were employed. Internodal interpolation has been achieved through the Weighted Upstream Differencing Scheme, which depends upon the computational Peclet number Pe . A study on the effect of the number of distinct grids on the overall algorithm performance is presented. Also reported is a comparative study of the computational effort as a function of Pe.

Keywords: Multigrid, Laminar flow, Convergence acceleration, Finite-volume, Malhas múltiplas, Escoamento laminar, Aceleração da convergência, Volumes finitos

COB1018 ANÁLISE DO DESEMPENHO DE DIFERENTES MÉTODOS PARA CALCULAR GRADIENTES EM MALHAS NÃO ESTRUTURADAS DE VORONOI / Analysis of performance of differents methods to calculate gradients in Voronoi unstructured mesh

Viviana Cocco Mariani & Sérgio Peters

Departamento de Informática e Estatística, Universidade Federal de Santa Catarina - UFSC - Florianópolis

CEP 88040-900 Florianópolis, Brasil - E-mail: viviana@inf.ufsc.br, peters@inf.ufsc.br

This paper presents an investigation on the solution of the linear systems arising from the numerical discretization of the Navier-Stokes equations through Voronoi grids. Three techniques are utilized and compared for evaluation of the pressure gradients: the minimum squared residue, the weighted mean between the projected normal gradients (MPGNP) and the modified weighted mean between the projected normal gradients, through a modification of the MPGNP. They are validated, adoting for the case of the incompressible flow in square cavity with sliding upper wall (driven cavity) with slide upper wall. All numerical results are obtained in distinct geometrical grids.

Keywords: Navier-Stokes, Voronoi grids, pressure gradients / Navier-Stokes, diagramas de Voronoi, gradientes de pressão.